In complex analysis

Complex analysis

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...

, Mittag-Leffler's theorem concerns the existence of meromorphic function

Meromorphic function

In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function...

s with prescribed poles. It is sister to the Weierstrass factorization theorem

Weierstrass factorization theorem

In mathematics, the Weierstrass factorization theorem in complex analysis, named after Karl Weierstrass, asserts that entire functions can be represented by a product involving their zeroes...

, which asserts existence of holomorphic function

Holomorphic function

In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...

s with prescribed zeros

Zero (complex analysis)

In complex analysis, a zero of a holomorphic function f is a complex number a such that f = 0.-Multiplicity of a zero:A complex number a is a simple zero of f, or a zero of multiplicity 1 of f, if f can be written asf=g\,where g is a holomorphic function g such that g is not zero.Generally, the...

. It is named after Gösta Mittag-Leffler.

Theorem

Let be an open set

Open set

The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...

 in and a closed

Closed set

In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...

 discrete subset. For each in , let be a polynomial in . There is a meromorphic function on such that for each , is holomorphic at . In particular, the principal part of at is .

One possible method of proof is as follows. Notice that if is finite, it suffices to take . If is not finite, consider the finite sum where is a finite subset of . Use Runge's theorem

Runge's theorem

In complex analysis, Runge's theorem, also known as Runge's approximation theorem, named after the German Mathematician Carl Runge, and put forward by him in the year 1885, states the following: If K is a compact subset of C , A is a set containing at least one complex number from every bounded...

 to construct a rational function approximating . As is exhausted by the finite sets , the approximations will approach a limit, which is the desired meromorphic function .

Example

Suppose that we desire a meromorphic function with simple poles of residue

Residue (complex analysis)

In mathematics, more specifically complex analysis, the residue is a complex number proportional to the contour integral of a meromorphic function along a path enclosing one of its singularities...

 1 at all positive integers. With notation as above, letting and , Mittag-Leffler's theorem asserts (non-constructively) the existence of a meromorphic function with principal part at for each positive integer . This has the desired properties. More constructively we can let . This series converges normally

Normal convergence

In mathematics normal convergence is a type of convergence for series of functions. Like absolute-convergence, it has the useful property that it is preserved when the order of summation is changed.- History :...

 on (as can be shown using the M-test) to a meromorphic function with the desired properties.